# std::norm(std::complex)

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< cpp‎ | numeric‎ | complex

Defined in header `<complex>` |  |
| (1) |
template< class T > T norm( const std::complex<T>& z ); |  | (until C++20)
template< class T > constexpr T norm( const std::complex<T>& z ); |  | (since C++20)
Additional overloads (since C++11) |  |
Defined in header `<complex>` |  |
| (A) |
float norm( float f ); double norm( double f ); long double norm( long double f ); |  | (until C++20)
constexpr float norm( float f ); constexpr double norm( double f ); constexpr long double norm( long double f ); |  | (since C++20) (until C++23)
template< class FloatingPoint > constexpr FloatingPoint norm( FloatingPoint f ); |  | (since C++23)
| (B) |
template< class Integer > double norm( Integer i ); |  | (until C++20)
template< class Integer > constexpr double norm( Integer i ); |  | (since C++20)

1) Returns the squared magnitude of the complex number z.

A,B) Additional overloads are provided for all integer and floating-point types, which are treated as complex numbers with zero imaginary component. | (since C++11)

### Parameters

z | - | complex value
f | - | floating-point value
i | - | integer value

### Return value

1) The squared magnitude of z.
A) The square of f.
B) The square of i.

### Notes

The norm calculated by this function is also known as field norm or absolute square.

The Euclidean norm of a complex number is provided by `std::abs`, which is more costly to compute. In some situations, it may be replaced by `std::norm`, for example, if abs(z1) > abs(z2) then norm(z1) > norm(z2).

The additional overloads are not required to be provided exactly as (A,B). They only need to be sufficient to ensure that for their argument num:

- If num has a standard(until C++23) floating-point type `T`, then std::norm(num) has the same effect as std::norm(std::complex<T>(num)).
- Otherwise, if num has an integer type, then std::norm(num) has the same effect as std::norm(std::complex<double>(num)).

### Example

Run this code

    #include <cassert>
    #include <complex>
    #include <iostream>

    int main()
    {
        constexpr std::complex<double> z {3.0, 4.0};
        static_assert(std::norm(z) == (z.real() * z.real() + z.imag() * z.imag()));
        static_assert(std::norm(z) == (z * std::conj(z)));
               assert(std::norm(z) == (std::abs(z) * std::abs(z)));
        std::cout << "std::norm(" << z << ") = " << std::norm(z) << '\n';
    }

Output:

    std::norm((3,4)) = 25

### See also

abs(std::complex) | returns the magnitude of a complex number (function template)
conj | returns the complex conjugate (function template)
polar | constructs a complex number from magnitude and phase angle (function template)

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